Definition 2 (Semialgebraic set and Semialgebraic function [1]}) 
A subset \(S\)  of \(\mathbb {R}^n\)  is called a real semialgebraic set if there exists a finite number of real polynomial functions \(P_{i,j}, \ Q_{i,j}: \mathbb {R}^n \rightarrow \mathbb {R}\) , such that
\(S = \bigcup _{j=1}^p \bigcap _{j=1}^q\lbrace  x \in \mathbb {R}^n: P_{i,j} =0, \ Q_{i,j} >0\rbrace .\) 
